Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...
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Exponential Map Computing
I am a Computer engineering student, and I am trying to implement the exponential map method on a discrete surface, from a tangent plane using vtk.
As a student I am facing some difficulties to ...
3
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1answer
44 views
Is the group of diffeomorphisms a Lie Group?
consider a smooth manifold and the group of diffeomorphisms (or (local) isometries in case of riemannian manifolds) $\varphi:M \rightarrow M$. How can one define a smooth structure on this group, s.t. ...
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16 views
Lie derivatives computation.
I know it is a bit of a localized question, but maybe somebody can give me a hint.
Let $X$ be a vector field and $\mathcal{L}_X$ the associated Lie operator. $\mathcal{L}_X^k$ is the operator which ...
4
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1answer
84 views
+50
Definition of smoothness “up to boundary”
Let $U\subseteq \mathbb{R}^n$ be an open set and let $f\in\mathcal{C}^k(U)$ for some positive integer $k$.
Are the following definitions of $\mathcal{C}^k$ regularity "up to boundary" equivalent?
...
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1answer
59 views
basis-independent isomorphism
Could some one give me some help with this proof? Given the hint, I still don't have a clue about how to proceed. Thanks.
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2answers
64 views
Question in do Carmo's book Riemannian geometry section 7
I have a question. Please help me.
Assume that $M$ is complete and noncompact, and let $p$ belong to $M$.
Show that $M$ contains a ray starting from $p$.
$M$ is a riemannian manifold. It is ...
2
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1answer
34 views
In a Frenet-Serret frame, what are $\Delta\vec T$ and $(\vec a\vec\nabla)\vec T$
Given a Frenet-Serret frame $(\vec T(t), \vec N(t), \vec B(t))$ defined by a curve $\vec \gamma(t)$ with
$$\begin{array}{rcl}
|\tfrac{d}{dt}\vec\gamma(t)| &\equiv& 1,
\\ \vec T(t) ...
3
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1answer
19 views
Please the Inequality in the proof of The Isoperimetric Inequality
From A proof of the Isoperimetric Inequality, can you please explain the starred inequality $$A + \pi r^2 = \int_{\gamma} x\,dy + \int_C -y\,dx = \int^l_0 x(s)y_s(s)\,ds - \int^l_0 ...
3
votes
2answers
40 views
Tangent map $F_{*}: T_{p}(M) \to T_{F(p)}(N)$ are linear tranformations.
How to show that tangent map $F_{*}: T_{p}(M) \to T_{F(p)}(N)$ are linear tranformations?
I know that it suffices to show that
$$F_{*}(ax_{u}+bx_{v})=aF_{*}(x_{u})+bF_{*}(x_{v})$$
where $x$ is a ...
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0answers
44 views
Integer conjugacy class
I am wondering about the following thing: What are integer conjugacy classes? Could anybody please give me a definition and maybe one or two examples?
What is meant with an integer conjugacy class ...
3
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0answers
27 views
Root Decomposition on Semisimple Lie Algebra over ${\bf C}$
Let $\mathfrak{g}$ is a semisimple Lie algebra over ${\bf C}$.
Then we have a direct sum $$ \mathfrak{g} = \mathfrak{h} + \sum_{\alpha}
\mathfrak{g}^\alpha $$.
where $\mathfrak{h}$ is a Cartan ...
2
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1answer
63 views
Calculating $ d \Phi^{*} \omega$ and $ \Phi^{*} d\omega$
Let $\omega \in \Omega^2(\mathbb{R}^3)$ as follow: $\omega = x dy\land dz + y dz \land dx + z dx \land dy $.
Let $\Phi: \Bbb R^3\to \Bbb R^3$ be given by
$$\Phi(r, \phi, \psi) = (r\cos\phi \cos\psi, ...
4
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0answers
62 views
approximating a variety locally by a vector space
Suppose we have $m$ homogeneous equations with integer coefficients in $n$ variables and that $m >> n$. Let $x_0 \in \mathbb{C}^n$.
Question 1: is there a way to approximate the variety ...
8
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1answer
95 views
Why is the Lie derivative linear in the vector field?
This might seem a very basic question, but I can't manage to find a proper proof in the books I have on my desk (or simply cannot see that it's "just that"). So be sure of what we talk about, let $G$ ...
1
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1answer
226 views
Helicoid and Catenoid
Let $X$ and $Y$ be isothermal parametrizations of minimal surfaces such that their component functions are pairwise harmonic conjugates, then $X$ and $Y$ are called conjugate minimal surfaces.
My ...
2
votes
1answer
27 views
For 1 form $\xi$, $F^{*}(d\xi)=d(F^{*}\xi)$
Let $F:M \to N$ be a mapping of surfaces, and $\xi$ be a function.
I want to show the following identity.
$$F^{*}(d\xi)=d(F^{*}\xi)$$
What I did :
Fix a tangent vector $v$.
...
1
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1answer
186 views
Definition of embedded and immersed curve
What does it mean to say that a curve in $\mathbb{R}^2$ is embedded? I think a curve in $\mathbb{R}^3$ is embedded if it lies on a plane, but what does it mean in 2d? I searched everywhere but I can't ...
1
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0answers
23 views
Orthochronous Lorentz is time preserving and SL(2,R)
Let's consider the pseudosphere in $\mathbb{R}^{1,2}$ given by $x^2+y^2-z^2=-R^2.$
We know that the group $O(1,2)=\{ A \in \operatorname{Mat}(3,\mathbb{R}): A^tGA=G \}$ where ...
5
votes
1answer
166 views
Embedded surface in $\mathbb{R}^3$
Let $U \subseteq \mathbb{R}^2$ be an open set and let $\sigma : U \rightarrow \mathbb{R}^3$ be a parametrization of an oriented surface $S$ embedded in $\mathbb{R}^3$ whose unit normal in $\sigma ...
4
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0answers
28 views
Dimension of the space of algebraic Riemann curvature tensors
Given $n\in \mathbb N$, consider the vector space $\mathbb R^{n^4}$ whose elements I will denote by $(R_{abcd})$ with indices $a,b,c,d \in \{1, \dots, n\}$. This vector space is $n^4$-dimensional. The ...
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votes
2answers
22 views
Regarding orientation and orientation-reversing in local diffeomorphism
I am confused about orientation and orientation reversing in local diffeomorphism $f$ from manifold $X$ to $Y$ at some points. So, what does $f$ orientation-reversing at a point mean?
5
votes
1answer
135 views
Where can I learn about complex differential forms?
So I'm a 3rd year grad student in number theory/modular forms/algebraic geometry, and I've worked with differential forms from an algebraic point of view without ever knowing what they really are. I'd ...
1
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1answer
41 views
property of the exterior derivative $d \circ d=$ for a $\mathcal C^\infty$ function
One of the properties of the exterior derivative is that $d\circ d=0$. We're trying to prove this for the case $f\in\mathcal C^\infty (U)$ on an open set $U\subset \mathbb R^n $.
The prove starts with ...
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0answers
21 views
focal surface conjugate
Could someone give me a prrof of Exercise 9 statement b from section 3-5 in "Ruled Surfaces and Minimal Surfaces of do Carmo's Differential Geometry of Curves and Surfaces" on page 211?
Link
Thanks
3
votes
1answer
59 views
Local Reparametrization of Surface using known Vector Field (Differential Geometry)
I need help with the following problem:
"Let $X$ be a vector field defined on surface $S$, and $p \in S$ such that $X(p) \neq 0$. Prove that there exists a local parametrization $\phi \colon U \to S$ ...
1
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1answer
59 views
Does nullity at one point implies nullity everywhere?
Consider the following definition of the derivation.
Now, consider a derivation $\delta_{p} :C^{r}(R^{n})\rightarrow R$ , i.e. it is defined on r-times differentiable functions defined on the ...
3
votes
1answer
39 views
When a smooth curve is an immersion (John Lee's Smooth manifold book p 156) and Example 7.3
In John Lee's Intro to Smooth Manifold book (2003 Springer) , I need some help with an example of an immersion.
On page 156 Example 7.1 c), If $\gamma(t): J \to M$ is a smooth curve ...then $\gamma$ ...
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1answer
30 views
Application of Uniformization Theorem
The statement is, if a compact, connected, orientable surface
has nonpositive Gaussian curvature, the Gauss–Bonnet theorem forces its
genus to be at least 1, and then the uniformization theorem tells ...
1
vote
1answer
36 views
Uniformization Theorem
I understand this statement of Uniformization Theorem:
Every simply connected Riemann surface is conformally
equivalent to the unit disk, the complex plane, or the Rieman $n$ sphere.
However, the ...
1
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0answers
38 views
How many normal planes?
Consider a surface $S$, a point $p$ on the surface, and the unit normal vector $\vec{N}$ passing through $p$ on the surface.
There are infinitely normal planes passing $p$ and its unit normal vector ...
2
votes
1answer
41 views
How many ways a surface can curve differently in different directions?
How many ways a surface can curve differently in different directions for a n-dimensional embedded submanifolds of $\mathbb{R}^m, m>n$? I think they can curve infinitely many ways but I am not ...
1
vote
2answers
53 views
manifold diffeomorphic (?) to SO(3)
Consider the set of all pairs $(\boldsymbol{n},\boldsymbol{v})$ of vectors in $\mathbb{R}^3$ such that $\boldsymbol{n}$ is a vector on the unit sphere centered at the origin and $\boldsymbol{v}$ is a ...
2
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2answers
129 views
Normal coordinates
Let $M$ a riemannian manifold and $\nabla$ the Levi-Civita conection. Ineed to prove the next.
Let $B$ an open ball of radius $r$ in $T_pM$ such that $exp_p\mid _B$ be a difeomorphism over an open $U$ ...
1
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2answers
58 views
How to define intrinsic curvature?
I have been exploring differential geometry slightly... And i'm trying to grasp how to define intrinsic curvature, from a visual/geometric viewpoint...
One formulation I got was that Intrinsic ...
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0answers
23 views
Geodesic Interpolation of a Vector
I have two vectors given and I want to estimate another vector by using geodesic interpolation, how can I do this? Thanks
2
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0answers
62 views
Manifolds: A definition of the Gradient an Algebraic Tangent vector over charts, how to show equivalence?
Let X be an n-dimensional differentiable manifold and $p \in X$ . Let $(U, h, V )$ for X around p with coordinates $(x_1 , . . . , x_n )$ in V , and let $v_i , i = 1, . . . , n$ , be the basis of ...
1
vote
2answers
55 views
closure of open set in topological space
How to prove : If $X$ is a topological space, $U$ is open in $X$, and $A$ is dense in $X$, then $\overline{U}=\overline{U \cap A}$
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votes
1answer
92 views
Tangent Vectors and Differential 1-forms.
I have this 1-form on $\mathbb R^3$ given by $\omega=dz+\frac{x}{2}dy-\frac{y}{2}dx$. If $p_0=(x_0,y_0,z_0)$ and $\vec v=(u_0,v_0,w_0)$, then find the set of tangent vectors $\vec v_{p_0}$ such that ...
1
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1answer
50 views
When to make a substitution in ODE
The setting is on evolving hypersurfaces. So for each time $t$, $\Gamma(t)$ is a hypersurface given by the zero level set of the function $\phi(x,t)$. Consider a ball, then the hypersurface has ...
2
votes
1answer
40 views
Smooth retraction onto a differentiable manifold
Let $M\subset\mathbb{R}^n$ be a smooth k-dimensional differentiable manifold (by which I mean that it is locally diffeomorphic to an open set in $\mathbb{R}^k$). Let us suppose $M$ compact for ...
3
votes
2answers
75 views
How can a group of matrices form a manifold?
So for example, $GL(n,\mathbb{R})$ group. It is said that this group can be considered as manifold - but I do not get how this is possible. How does one then assign a neighborhood of a matrix, and ...
6
votes
2answers
81 views
Are there any simply connected parallelizable 4-manifolds?
On pp. 166 of Scorpan's "The Wild World of 4--manifolds", he gives an example of a parallelizable 4-manifold ($S^1\times S^3$) and then asserts: "there are no simply connected examples". It's ...
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1answer
16 views
How does one prove that local diffeomorphism is submersion?
How does one prove that local diffeomorphism is submersion?
For a manifold, what does it being disconnected mean? I get what "disconnected" means for a graph, but not for a manifold.
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1answer
26 views
What form of Leibniz rule is this (principal fiber bundle)?
Let $P(M,G)$ be a principal fiber bundle. Let $\sigma : U \subseteq M \rightarrow P$ be a smooth local section and $f : U \rightarrow G$ a smooth function. For $ a \in G$, $R_a : P \rightarrow P$ is ...
2
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0answers
39 views
What is the torsion and curvature in modern language?
I think it can not be the torsion and curvature in the connection context, for these two are anti-symmetric in two variables, so that they must vanish on a one-dimensional space (tangent space of a ...
1
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0answers
26 views
On ruled surfaces
Let $E,F,G,L,M,N$ are the the coefficients of the first and second fundamental forms of a surface. How can I say that the surface is a ruled surface or not? (Only with using these coefficients)
1
vote
0answers
26 views
Asymptotic invariants of infinite groups
I am reading a Gromov's book " Metric Structures for Riemannian and Non-Riemannian Spaces ". Consider the following concept :
$$ distort(X)\doteq sup \frac{length\ dist|_X}{dist|_X} $$
That is for ...
5
votes
1answer
43 views
Concerning the tangent space of an exotic $\mathbb R^4$
My geometric intuition is very poor, so my naive approach to this question is "if $M$ is an exotic $\mathbb R^4$, then $TM$ must be something like $\mathbb R^8$, which is not exotic". Of course, my ...
3
votes
2answers
48 views
Showing that $E=X/{\sim}$ (where $X=[0,1]\times\mathbb{R}^n$) is a smooth vector bundle over $S^1$
I want to show that $E=X/{\sim}$ (where $X=[0,1]\times\mathbb{R}^n$) is the total space of a smooth vector bundle over $S^1$. Here $\sim$ is defined as follows: fix a linear isomorphism $L$ of ...
1
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1answer
38 views
uniformization theorem - squares and circles
I am trying to understand the uniformization theorem and get some intuition about it.
The uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of ...
